#495842

order	512
nilpotency class	4
isoclinism	isoclinic class of #483882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(2,4)(5,6,10,15)(7,12,18,33)(8,21)(9,26,28,54)(11,31,30,16)(13,36)(14,40,42,29)(17,43)(19,49,47,65)(20,52,48,35)(22,55)(23,39)(24,41)(25,56)(27,37)(32,58)(34,62,61,51)(38,57)(44,63)(45,60)(46,67)(50,64)(53,66)(59,68);(1,2,3,4)(5,7,10,18)(6,12,15,33)(8,17,25,46)(9,19,28,47)(11,20,30,48)(13,32,39,60)(14,34,42,61)(16,35,31,52)(21,43,56,67)(22,53,24,44)(23,45,36,58)(26,49,54,65)(27,50,57,68)(29,51,40,62)(37,64,38,59)(41,63,55,66);(5,8)(6,13)(7,17)(9,27)(10,25)(11,22)(12,32)(14,41)(15,39)(16,37)(18,46)(19,50)(20,53)(21,54)(23,40)(24,30)(26,56)(28,57)(29,36)(31,38)(33,60)(34,63)(35,64)(42,55)(43,65)(44,48)(45,62)(47,68)(49,67)(51,58)(52,59)(61,66);(7,18)(8,22,25,24)(9,28)(12,33)(13,37,39,38)(14,42)(17,44,46,53)(20,48)(21,55,56,41)(23,57,36,27)(26,54)(29,40)(32,59,60,64)(35,52)(43,63,67,66)(45,50,58,68)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=gi,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,d^c=dh,f^c=fhi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343336, #72343342, #72343476
maximal normal subgroup of	#72343336, #72343342, #72343476

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403038);